Study on knots by diagrammatic approach

• Project/Area Number: 19K03492
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Kobe University
• Principal Investigator: Nakanishi Yasutaka 神戸大学, 理学研究科, 名誉教授 (70183514)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Project Status: Completed (Fiscal Year 2023)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2021: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2020: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2019: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 結び目 / 図式的アプローチ / 局所変形

Outline of Research at the Start

結び目は豊かな構造をもつ対象である。近年、結び目不変量の開発と研究が進み、様々な解釈やアプローチが可能になってきている。不変量の幾何学的な意味合いと特徴づけ、豊かな構造のどの側面が反映されているのかを究明したい。
本研究の具体的な目的は、結び目の構造と不変量を図式的アプローチを通じて明らかにし、その発展として、結び目空間の組み合わせ的構造を究明することである。
上記の研究目的の達成するために、「図式的アプローチの開発」と「不変量の究明」に重点を置いて研究を進める。こうした基盤のもとに、「仮想結び目への応用」ならびに「彩色数への応用」に取り組んでいく。

Outline of Final Research Achievements

We have studied structures and invariants of knots by local moves to research the structure of knot space. We give a local move corresponding to writhe polynomial, which is a fundamental invariant of virtual knots. By considering this result, we give intersection polynomials, which are new invariants of virtual knots. Furthermore, we have studied their characters.

Academic Significance and Societal Importance of the Research Achievements

研究成果により結び目理論における新たな知見を与えることができた。この成果は結び目理論を通じて、位相幾何学ならびに数学の発展に寄与するものだと判断する。また、結び目の局所変形を通じて、DNA 結び目や高分子結び目の研究にも応用されることを期待している。

Research Products

• [Journal Article] The intersection polynomials of a virtual knot III: Characterization (2024)
  • Author(s): Ryuji Higa, Takuji Nakamura, Yasutaka Nakanishi, and Shin Satoh
  • Journal Title: Osaka Journal of Mathematics Volume: 61 Pages: 229-245
  • Peer Reviewed / Open Access
• [Journal Article] The intersection polynomials of a virtual knot I: Definitions and calculations (2023)
  • Author(s): Higa Rayuji、Nakamura Takuji、Nakanishi Yasutaka、Satoh Shin
  • Journal Title: Indiana University Mathematics Journal Volume: 72 Issue: 6 Pages: 2369-2401
  • DOI: 10.1512/iumj.2023.72.9599
  • Peer Reviewed / Open Access
• [Journal Article] The intersection polynomials of a virtual knot II: Connected sums (2023)
  • Author(s): Higa Ryuji、Nakamura Takuji、Nakanishi Yasutaka、Satoh Shin
  • Journal Title: Journal of Knot Theory and Its Ramifications Volume: 32 Issue: 10 Pages: 2350067-2350067
  • DOI: 10.1142/s0218216523500670
  • Peer Reviewed
• [Journal Article] On pretzel links which are concordant to the trivial link (2023)
  • Author(s): Nakanishi Yasutaka、Shibuya Tetsuo、Tsukamoto Tatsuya
  • Journal Title: Journal of Knot Theory and Its Ramifications Volume: 32 Issue: 13 Pages: 2350083-2350083
  • DOI: 10.1142/s0218216523500839
  • Peer Reviewed
• [Journal Article] The intersection polynomials of a virtual knot III: Characterization (2023)
  • Author(s): Ryuji Higa, Takuji Nakamura, Yasutaka Nakanishi, and Shin Satoh
  • Journal Title: Osaka Journal of Mathematics Volume: -
  • Peer Reviewed
• [Journal Article] Writhe polynomials and shell moves for virtual knots and links (2020)
  • Author(s): Nakamura Takuji, Nakanishi Yasutaka, Satoh Shin
  • Journal Title: European Journal of Combinatorics Volume: 84 Pages: 103033-103033
  • DOI: 10.1016/j.ejc.2019.103033
  • Peer Reviewed / Open Access
• [Journal Article] A note on coverings of virtual knots (2019)
  • Author(s): Nakamura Takuji, Nakanishi Yasutaka, Satoh Shin
  • Journal Title: Journal of Knot Theory and its Ramifications Volume: Online Ready Issue: 08 Pages: 1971002-1971002
  • DOI: 10.1142/s0218216519710020
  • Peer Reviewed / Open Access
• [Presentation] Colors of virtual tangles (2023)
  • Author(s): 中西康剛, 佐藤進
  • Organizer: 日本数学会トポロジー分科会
• [Presentation] The differences of Conway polynomials for knots caused by a single pass move (2022)
  • Author(s): 中西康剛, 高木駿希
  • Organizer: 研究集会「拡大KOOKセミナー2022」
  • Invited
• [Presentation] The differences of Conway polynomials for knots caused by a single pass move (2022)
  • Author(s): 中西康剛, 高木駿希
  • Organizer: 日本数学会・トポロジー分科会
• [Presentation] The intersection polynomials of a virtual knot (2021)
  • Author(s): Yasutaka Nakanishi
  • Organizer: 12-th TAPU-KOOK Joint Seminar on Knots and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] Alexander polynomials and crossing changes (2020)
  • Author(s): 中西康剛
  • Organizer: 2019 琉球結び目セミナー
  • Invited
• [Presentation] A note on sharp move (2019)
  • Author(s): Yasutaka Nakanishi
  • Organizer: International Workshop "Knots in Tsushima 2019"
  • Int'l Joint Research / Invited